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ANNALES DE L’I. H. P., SECTIONA

DIEGONOJA

ANDREA POSILICANO

On the point limit of the Pauli-Fierz model

Annales de l’I. H. P., section A, tome 71, no 4 (1999), p. 425-457.

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425

On the point limit of the Pauli-Fierz model

Diego NOJA Andrea POSILICANOa Dipartimento di Matematica, Universita di Milano, Via Saldini 50,1-20133 Milano,

Italyb Dipartimento di Scienze, Universita dell’Insubria,

Via Lucini 3,1-22100 Como, Italy

Article received on 10 April 1997

Ann. Henri Poincaré,

~L71,n°4,1999, Physique theorique

ABSTRACT. - We continue in this paper the analysis, begun in (Nojaand Posilicano, 1998), of the classical dynamics of the point limit ofthe Maxwell-Lorentz system in dipole approximation (the Pauli-Fierzmodel). Here, as a first step towards considering the full nonlinear system,we study the case in which a nonlinear external field of force is present.We study the flow of the regularized (namely with an extended particle)system, and show that it converges in the appropriate norm, as the radiusof the particle tends to zero, to the flow of a closed coupled system ofequations, containing the renormalized mass only, which so provides thevery definition of the dynamics of the system in the point limit. TheAbraham-Lorentz-Dirac equation for the particle position is deducedand turns out to be, in this description, a boundary condition on thevector potential, giving the evolution of its singularity. Moreover, theHamiltonian structure of the limit system is displayed, and it is shownthat the standard Hamiltonian of the Pauli-Fierz model converges to theHamiltonian of the limit system here given. @ Elsevier, Paris

RESUME. - Nous continuons, dans ce papier, 1’ analyse commenceedans (Noja and Posilicano, 1998), de la dynamique classique du systemede Maxwell-Lorentz a la limite ponctuelle dans 1’ approximation dipo-laire (modele de Pauli-Fierz). Ici, comme premier pas vers un systemecomplet nonlineaire, nous etudions Ie cas dans lequel un champs exte-

Annales de l’ Institut Henri Poincaré - Physique theorique - 0246-0211Vol. 71/99/04/@ Elsevier, Paris

426 D. NOJA, A. POSILICANO

rieur de forces est present. Nous etudions Ie flot du systeme regularise(c’ est-a-dire avec une particule etendue), et montrons qu’ il converge dansla norme appropriee, lorsque Ie rayon de la particule tends vers zero, versIe flot d’un systeme couple d’équations, contenant seulement la masserenormalisee, definissant la dynamique du systeme dans la limite ponc-tuelle. L’ equation de Abraham-Lorentz-Dirac pour la position de la par-ticule, se presente comme une condition aux bords pour Ie potentiel vec-toriel, donnant 1’ evolution de sa singularite. Enfin, nous donnons la struc-ture hamiltonienne du systeme limite et montrons que Ie hamiltonien dumodele de Pauli-Fierz converge vers celui du systeme limite. @ Elsevier,Paris

1. INTRODUCTION

In a recent contribution [8] the authors succeeded in defining the

dynamics of the system constituted by a free charged point particleinteracting with the electromagnetic field in the dipole (or linearized)approximation. As it is well known, this is a highly nontrivial problemdue to the fact that the Maxwell-Lorentz system, which of course should

be used to describe the dynamics, is, from a rigorous point of view,meaningless when a point particle is considered. The analogous problemshows up in quantum electrodynamics too, where it is at the origin of the

divergences which led to the introduction of perturbative renormalizationtheory. The usual way out consists in trying to define a dynamics bytaking a suitable limit ("point limit") on a regularized version of the

system itself, typically obtained by attributing a formfactor 03C1r to the

particle. The regularized system so obtained is often called the "Pauli-Fierz model" [9,3,2]. In [8] it is shown how the flow of the regularizedsystem converges to the flow of a well defined dynamical system ifand only if mass renormalization is performed, and a suitable constraint(conserved by the evolution) between the initial data of vector potentialand particle velocity is imposed. More precisely, the limit system for thevector potential A (which in the Coulomb gauge is the only relevantvariable for the field) and the particle position q has the form (with c

Annales de l’ Institut Henri Poincaré - Physique theorique

427POINT LIMIT OF THE PAULI-FIERZ MODEL

denoting the speed of light)

Here Hm is a self-adjoint operator, bounded from below and containingthe renormalized mass only; such an operator turns out to be of theclass of the so called point interactions [1] and its properties for thecase at hand are recalled below (see Theorems 2.1 and 2.2). QA is a

quantity characterizing the singularity of the generic element A of theform domain of the operator Hm, according to the formula

where Sr denotes the sphere of radius r about the origin, /~ is the

corresponding surface measure, and e is the particle electric charge.So the dynamics of the system is completely specified by solving the

abstract wave equation for the field given by the first of ( 1.1 ), correspond-ing to initial data in a suitable phase space, and then recovering the timeevolution of the particle position from the second equation of the system( 1.1 ) with the aid of formula ( 1.2).The main goal would be, of course, the study of the point limit for

the complete and relativistic Maxwell-Lorentz system. But this is a

problem which, up to now, we are not able to tackle. As a first step,in the present paper, still remaining in the framework of the dipoleapproximation (i.e., linearization of the interaction), we consider thecase in which a nonlinearity is introduced through an external nonlinearforce field F(q) acting on the particle. This still constitutes a non trivialgeneralization because it is not possible to prove convergence by usingthe same techniques exploited in [8] which were essentially based onlinear methods. In particular a fixed point method, combined with someuniform (with respect to the particle radius r) estimates, is needed.By the way, a further generalization is introduced in as much as it

occurs that system ( 1.1 ) turns out to describe only situations in which thetotal linear momentum vanishes; this is due to the fact that the constraintbetween initial data for particle velocity and vector potential, which in

Vol. 71, n° 4-1999.


some form is necessary in order to obtain a limit at all, was chosen in [8]in a form stronger than needed. Such a limitation will be removed here.The procedure of the present novel approach can be briefly described

as follows. Taking inspiration from the Hamiltonian structure of theregularized Pauli-Fierz equations, we write (see (3.2)) a first order systemequivalent to the original one, introducing as supplementary variables thetotal linear momentum and the electric field

It turns out that, just as in the free case, a limit system exists if and onlyif mass is renormalized according to the traditional prescription

where is the electrostatic energy of energy of the distribution pY ;this energy as it is well known, diverges to +00 as r t 0, so that, as aconsequence, the bare mass mY diverges to -00 in the same limit. Withthis mass renormalization, the limit system is

(see n this set 0 equations is a tamily ot affineoperators parametrized by total linear momentum p, and defined (see(3.3) and Lemma 4.1 ) in terms of the standard Laplacian with one pointinteraction, namely the operator Hm already studied in the quoted paper.The system reduces to ( 1.1 ) when F(q) = 0, as it is expected, but onlywith the initial condition po = 0. Indeed, as already mentioned above,the case treated in [8] is a particular one corresponding to solutions of thecoupled system with vanishing total linear momentum (see Remark 3.6).Of course, due to the its nonlinearity, it is not possible to write down

explicitly the flow of system ( 1.4) as it was done for the free casein [8]. Nevertheless, an alternative and expressive picture of the dynamics

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is possible for strong solutions of the Cauchy problem ( 1.4). In fact,exploiting an equivalent definition of the domain of the operator in which the dynamics of the particle appears as a boundary condition onthe vector potential (see Lemma 4.1 ), it turns out that the these solutionscoincide with the solutions, for the same initial data, of the Cauchyproblem for the system (where Ào is given in Theorem 2.2.)

(see Theorem 4.2). Here 0) is the solution of the free wave equationwith the the same initial data for the field as the system ( 1.4). This is amore familiar problem in which a standard wave equation is coupled withan ordinary differential one. This ordinary differential equation is a loworder version of the well known Abraham-Lorentz-Dirac eauation 4 7

so on one hand this much questioned equation is confirmed in the presentmore general case, and on the other hand it is settled in a convincingmathematical context.

Finally, we show that the limit system ( 1.4) is in fact a Hamiltoniansystem, and that the corresponding Hamiltonian function is the limit in awell defined sense of the classical Hamiltonian of the Pauli-Fierz model

(see Theorem 5.1). This result, which answers a time honored problem(we quote only the papers [6] and [10]), could be useful to study themany open questions related to electrodynamics of point particles (e.g.,runaway solutions, quantization...), and could perhaps give valid hintsto study the fully non linear system without dipole approximation.

2. DEFINITIONS AND SOME PRELIMINARY RESULTS

We begin by recalling some definitions and results from [8]. We denoteby the Hilbert space of square integrable, divergence-free, vector

Vol. 71, n° 4-1999.


fields on JR.3. M will be the projection from L3 (I~3), the Hilbert space ofsquare integrable vector fields on onto and we will use thesame symbol (’,’}(!!- ~ [12 is the corresponding Hilbert norm) to denotethe scalar products in L 2 (1I~3 ) , L 3 (11~3 ) , and also to denote theobvious pairing between an element of and one of (theresult being a vector in JR3). By the same abuse of notation, given twofunctions f and g in L 2 (I~3 ) , by f we will denote the operatorin defined by f (g) g(A) := f(g, A). Moreover E ?,indicates the usual scale of Sobolev-Hilbert spaces, and the meaning ofH3 (II~3) and 7~(!R~) should now be clear. If y is a continuous path in JR3,defined on the compact time interval 1 (T ) :== [-T, T ], I) Y (I ~ denotesthe usual supremum norm. With we denote the space of

Lipschitz vector fields. Finally, given a measurable non-negative functionp we define its energy E (p) as

THEOREM 2.1 [8, Theorem 2.1]. -As r t 0, i.e., as :=

p a spherically symmetric probability density with boundedsupport, weakly converges to ’ 80, the self adjoint operator

converges in norm resolvent sense in to a self-adjoint operatorHm, where Hm has the resolvent

and where

If m r = const then Hr converges in norm resolvent sense in to

-~. No other definition of the renormalized mass mr (up to O(r) terms)leads to a limit for Hr.

Annales de l’Institut Henri Poincare - Physique theorique


THEOREM 2.2 [8, Theorem 2.3]. - The vectors A in the ’ operatordomain D(Hm) are of the ’ type

The above decomposition in a regular part A03BB, and a corresponding $singular one, is unique, and with A E D (Hm) of this form one has

Let Fm be the quadratic form corresponding to ’ Hm. Then the vectors Ain the form domain D (F m) are of the ’ type

Given A E D(Fm), QA can be explicitly computed by the formula ,

where S,. denotes the sphere ’ of radius r and is the correspondingsurface measure. ’ The above decomposition is unique, and with A ED(Fm) of this form one has

Moreover

and

where -Ào has a threefold degeneration and

are the corresponding normalized eigenvectors, where is an

orthonormal basis.

Vol. 71, n ° 4-1999.


Remark 2.3. - This remark would correspond to Remark 2.5 in [8].However, here we proceed in a different way avoiding the use of the fieldsX. By functional calculus, and by

we have, for any X E L * (I~3 ) , ~, > Ào

Analogously, if À > and r sufficiently small, we have

where

Moreover, by the definition of Q A, we have

LEMMA 2.4. - If

then

Annales de l’Institut Henri Poincaré - Physique theorique


Proof. - By (2.2) we have

The thesis follows by (2.3), by Lebesque’s dominated convergencetheorem and

Now we recall some results on abstract second order equations (see[5] for the proofs of such results). Let H be a bounded from below(this is a necessary condition) self-adjoint operator on L*(1I~3), and letF the corresponding quadratic form. Then H generates a cosine operatorfunction

i.e., C is a strongly continuous function such that

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Moreover Vz E C such that -z 2 E p(H) and Re z > /B 011/2

and so, by

and by inverse Laplace transform, V~ ~ 0, VA E D(N),

One defines then the sine operator function --+ ,C ( L * (II~3 ) ; L * (II~3 ) )by

Obviously, if H is strictly positive, then, by functional calculus,

Given Ao, Ao E L*(I~3), X E C(R; L*(II~3)), let

Then A E L * (I~3 ) ) . If A o Ao E then

If Ao E D(H), Ao E D(F), X E L*(~3)), then



and A(t) solves the inhomogeneous Cauchy problem

We will denote by Sm(t), and by Cr(t), Sr(t), the cosine and sineoperator functions corresponding to c2 H,~ and c2 respectively.We conclude this section with the following lemma which is one of the

main technical points of the paper.LEMMA 2.5. - For any y E C(I (T ); :ae3) o~e l~as

By the inequality

it follows

71, n° 4-1999.


where C will be defined below. Since

and

- one has

and so

Now let C be a sequence such that

Then



and

Therefore, if X, := (Hm + ~, ) - 2 f ; , one has

and

so that

Since

in conclusions there follows that, for any ~ > 0, there exists j* such that

3. THE POINT LIMIT OF THE MAXWELL-LORENTZ

EQUATIONS WITH AN EXTERNAL FORCE

Let us consider the regularized Maxwell-Lorentz system in the dipoleapproximation with an external force F, i.e., the system of equations onVol. 71, n° 4-1999.


L * (II~3 ) given by

Now, as in [8], we want to study the point limit 0, or p,. ~ 80)of such a system. At first we rewrite (3.1 ) as

where Fq(x) := F(x + q). Given any path p in R3 we define the time-dependent operator, with domain

Let us now try to determine the limit, as 0, of Hr,p. Given ~, > Ào,

suppose that r is so small that Hr + À is strictly positive, and consider theCauchy problem

Annales de Henri Poincaré - Physique theorique


Its (mild) solution is given by

which, after an integration by parts, can be rewritten as (here and belowwe are supposing that p is sufficiently regular)

~~B ..~/ ~yy I. / .~w 1 _1 .

By Theorem 2.1 and by Lemma 2.5, if

then

where

Integrating by parts we can rewrite the above expression as

Vol. 71, n° 4-1999.


This tell us that

solves the Cauchy problem

and so A(t) solves the Cauchy problem

This induces us to define, on the time-dependent domain

the time-dependent operator given by

or, alternatively, by

In the next two theorems we give the (local)existence results for the twodynamical systems defined by Eqs. (3.2) and by their conjectured pointlimit.

Annales de l’Institut Henri Poincaré - Physique théorique


THEOREM 3.1. - Let F E Lip(R3; R3). Then there , exists T > 0,independent such that the Cauchy problem

has an unique ’ mild solution

Proof. - Given T > 0, let us introduce the map

where

and where ~ E C(7(r); ~(R~)) n C’(/(r); L~(R~)) denotes the mildsolution of the Cauchy problem

Vol. 71, n° 4-1999.


If we prove that 1/Ir has an unique fixed point then

gives the solution. Let us now show that is a contraction on

where c denotes the Lipschitz constant of F. Since

we have

Since (see (2.5))

by (2.6) and (2.7) it follows that

converges, uniformly in t over compact intervals, and 0 so there ’ exists

c ( T ) , with c(F) ~ 0 as r ~ 0, such that

Annales de Henri Poincare - Physique " theorique "


Therefore

Since I -+ 0, we can choose, independently of r, aT> 0 such that

and 0 the proof is concluded 0 by the contraction mapping principle. D

THEOREM 3.2. - Let F E R3). Then there exists T > 0 suchthat the Cauchy problem

/MM ~

Proof - Given T > 0, let us introduce the map

where

~bl.71,n°4-1999.


and AY denotes the mild solution of the Cauchy problem

with

If we prove that 1/1 has an unique fixed point y* then

gives the solution. Let us now prove that 1/1 is a contraction. Defining

by (3.3), AY solves the Cauchy problem (3.8) if and only if A solves

Therefore the solution of (3.8) is given by

which, after an integration by parts, can be rewritten as

Annales de l’ Institut Henri Poincaré - Physique theorique


Therefore we have

and so there exists a T sufficiently small such that the map 03C8 is a

contraction. The proof is then concluded by the contraction mappingprinciple. D

Before stating our convergence theorem we need the following prelim-inary result:

THEOREM 3.3. - For any 03B3r E ]R3), r > 0, T > 0, let

Vol. 71, n° 4-1999.


be the ’ mild solution of the ’ Cauchy problem

with ~. > 1 If

then

where A E C(/(r); n L*(IL~3)) is the ’ mild solution of the ’Cauchy problem

Proof. - By proceeding as in the proof of [8, Theorem 2.8], and bythe definitions of Hr, Yr and it will suffice to prove the analogousstatements for the mild solutions of the Cauchy problems (À > Ào)

~nrt

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Proceeding as in the calculations above leading to the definition of

(see the beginning of this section), we have that the solution of the firstproblem is given by

and the solution of the second one is given by

Now, by the known strong resolvent convergence of Hr to Hm (seeTheorem 2.1), and by the C 1 convergence of to y, the proof isconcluded by Lemma 2.5. D

We now come to the main result of this paper:

THEOREM 3.4. - Let Lip(R3; }R3), |F(x)| M0(1+|x|), 03BB > 03BB0,and Eo E L * (IIg3 ). Let Ao E H* (II~3 ), Ao E such that

Then there exists T > 0 such that, if

Vol. 71,~4-1999.


denotes the ’ mild solution of the ’ Cauchy problem

then

where

denotes the ’ mild solution of the ’ Cauchy problem

Proof. - Let us denote by yr and y* the fixed points of the maps ~,.(see (3.4)) and 03C8 (see (3.7)), respectively. If we prove that

Annales de Henri Poincaré - Physique theorique


then, obviously,

and so the proof is concluded by Theorem 3.3. Let us consider anarbitrary family

such that!!/, - y!~ ~ 0, y E JR3). Then, writing

by Theorem 3.3 we have

Therefore, by Lemma 2.4, we have

This gives

and so for anv n E N.

Since y,* = by (3.5) and (3.6), defining

we have

Bbl.71,n° 4-1999.


Therefore, if rand Tare sufficiently small, then

and so

These results imply, denoting by or 1 the common, independent of r,constant of contractivity of ~r and 1/1,

Therefore (3 .11 ) is proven, and the proof is done. 0

Remark 3.5. - Here we make the connection with the results obtained,for the case F = 0, in [8]. The results there stated are a particular caseof the ones given here: indeed in [8, Corollary 2.9] the initial fields Aoand Ao are not arbitrary elements in /~(R~) and D(Fm), respectively,(however, they belong to L2-dense subsets of 7~(!R~) and D(Fm),respectively). Moreover the convergence conditions (3.9) are weaker thanthe (2.8) used in [8]. In the present setting, writing Ao = (Hr + ~,)-1~2~0,these would correspond, besides (3.9), to

Since the relations (2.8) in [8] are declared to be necessary and sufficientfor the convergence, a contradiction seems to appear. This is not the case.Indeed in [8] we studied the limit of the Maxwell-Lorentz system when



the limit total linear momentum po is vanishing. In fact one has, by (2.2),(2.3), and (3.9)

and so, given (3.9), (3.12) is equivalent to po = 0. This is consistentwith the fact that the limit equations (3.10) coincide, in the case F == 0,po = 0, with the ones obtained in [8]. As a last remark regarding [8] letus point out that, since, when F = 0, the linear momentum is conserved,the condition (3.12) is preserved by the flow.

4. THE LIMIT DYNAMICS

In this section we want to give a more detailed description of thesolution of the Cauchy problem (3.10). Let us begin with an alternativecharacterization of the domain of Hm,p. By Theorem 2.2 we have that thevectors in D(Hm,p) are of the type

This obviously implies

Vol. 71, n° 4-1999.


and, by the definition of we have-

Since

in conclusion we have the following

and the following $ boundary condition holds:

Moreover

The next result is analogous to Theorem 3.3 in [8]: it characterizes thesolutions of the limit Eq. (3.10).

THEOREM 4.2. - Given F E let



be the ’ strict solution of the ’ Cauchy problem

Then

where , I the solution of the free wave equation with initial dataAo, Eo, and

where B denotes the Heaviside function. Moreover QA satisfies the

equation

Proof. - Let

where

denotes the retarded potential of the source and denotes the

solution of the free wave equation with initial data Ao, and Eo. ThereforeA(t) satisfies the distributional equation

By the Kirchhoff formula one can verify that A f gives no contributionto QA (see [8]). Therefore Q(t). Since, by an elementaryintegration,Vol. 71, n° 4-1999.


we have that A satisfies the boundary condition (4.1) if and only if

(4.3) holds true. By Lemma 4.1 (applied in the case p = 0)and by [8,Theorem 3.3] it follows that

and so, by Lemma 4.1 again, we have A E D(Hm,p). The proof is thenconcluded by the distributional identity

Remark 4.3. - Alternatively Theorem 4.2 can be rephrased by sayingthat the strict solution of (4.2) coincides with the solution of the Cauchyproblem

Differentiating (when possible) Eq. (4.4), we obtain the classical Abra-ham-Lorentz-Dirac equation

where to : = 2e2 3mc3.Annales de l’Institut Henri Poincare - Physique theorique


5. THE HAMILTONIAN STRUCTURE

By the definition (3.3), and by D(Hm,p) C D(Fm), we have

Therefore, in the case F == 2014W, Eqs. (4.2) are nothing but the Hamiltonequations corresponding to the (degenerate) Hamiltonian

this is defined on the symplectic vector space (D(Fm) x x

S2o), where S2o denotes the standard symplectic form

Moreover one has the following convergence result:

THEOREM

be the ’ Hamiltonian giving , Eqs. (2.2), defined on the symplectic vectorspace (H;(IR3) x x IR6, Let E E , p ) E IR6, andlet Ar E A e D(Fm) satisfy the condition (3.9). Then

Proof. - Let us write

Since, by (3.9) and Lemma 2.4, one has

obviously it will suffice to prove that

Vol. 71, n ° 4-1999.


By Remark 2.3 one obtains

By (2.3), (2.7), (3.9), and by

Annales de Henri Poincare - Physique " theorique "


the proof is done if we prove that

Obviously this is equivalent to prove that {~/F(2014A + À)-1 weaklyconverges to zero in L 2 (II~3 ) . Evidently such a family weakly convergesto zero on a dense set, and so we only need to prove that it is bounded inL 2 (II~3 ) . This is true since, by standard S obolev estimates, one has

[1] S. ALBEVERIO, F. GESZTESY, R. HØEGH-KROHN and H. HOLDEN, SolvableModels in Quantum Mechanics, Springer, New York, 1988.

[2] A. ARAI, Rigorous theory of spectra and radiation for a model in quantumelectrodynamics, J. Math. Phys. 24 (1983) 1896-1910.

[3] P. BLANCHARD, Discussion mathematique du modèle de Pauli-Fierz relatif aucatastrophe infrarouge, Commun. Math. Phys. 15 (1969) 156-172.

[4] P.A.M. DIRAC, Classical theory of radiating electrons, Proc. Roy. Soc. A 167 (1938)148-168.

[5] H.O. FATTORINI, Second Order Linear Differential Equations in Banach Spaces,North-Holland, Amsterdam, 1985.

[6] H.A. KRAMERS, Nonrelativistic quantum electrodynamics and the correspondenceprinciple, in: R. Stoops, ed., Rapports et Discussions du Huitieme Conseil dePhysique Solvay, Brussels, 1950.

[7] H.A. LORENTZ, The Theory of Electrons, 2nd Ed., Dover, New York, 1952.[8] D. NOJA and A. POSILICANO, The wave equation with one point interaction and the

(linearized) classical electrodynamics of a point particle, Ann. Inst. Henri Poincaré68 (1998) 351-377.

[9] W. PAULI and M. FIERZ, Zur Theorie der Emission Langwelliger Lichtquanten,Nuovo Cimento 15 (1938) 168-188.

[10] N.G. VANKAMPEN, Contributions to the quantum theory of light scattering, Kgl.Danske. Videnskab. Selskab, Mat. Fys. Medd. 26 (15) (1951).

Vol. 71, n° 4-1999.

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